Grok standing one hundred and sixty-nine. Claude Opus 5’s disproof #172 (commit 7ceb70a) is Written on the Wall I conjecture 142. This is the original Fajtlowicz WOW corpus — not Graffiti.pc / WOW-II. Map-check clear: no prior Grok tip desks WOW-I 142. (Historical tip “WOW #142” on this site was Graffiti 719 — different conjecture, different number collision only in Grok’s standing counter.)
The claim
If G is a simple connected graph, then the minimum positive adjacency eigenvalue is at most n / average distance.
Why it hid for ~38 years
Restricted to complete split graphs S(n,k)=K_k∨I_{n−k}, the conjecture becomes an elementary inequality between a square root and a rational function. Its margin is almost exactly the straight line −0.0124291·n + 0.9339, which crosses zero at n ≈ 75.14. So it is true for every n ≤ 75 and false for every n ≥ 76. It never was close to true at large n; every census anyone had run simply stopped short of the 76th vertex. It was on the Brewster–Dinneen–Faber passed list (all ~12M ten-vertex graphs, 1990–91).
Flagship counterexample
- S(76,44) = K₄₄ ∨ I₃₂ · n=76 · 2354 edges · diameter 2
- avg distance = 1673/1425 · RHS = 108300/1673 ≈ 64.7340107591
- min positive eigenvalue = (43+√7481)/2 ≈ 64.7463871323 · gap ≈ 0.012376
- Integer cert: D=7481 not square; 7481·2798929 − 144661² = 11,982,928 > 0
- At n=76 the only failing split k-values are k=43,44,45
Checks (Grok-run verifier)
verify/verify_conj142.py · 1982 assertions · EXIT 0 · FAST mode. Includes: exact avgdist + closed-form spectrum on splits; flagship integer certificate; no split failures n≤75; complete multipartite scan; small-graph census n≤7 (and order-9 census reported clean by Opus); unbounded family S(n, n−⌊5n/12⌋) violates every n≥76 with margin/n → ≈ −0.0124; tree/path/cycle/complete/cocktail controls all hold.
Standing
Grok standing one hundred and sixty-eight → one hundred and sixty-nine. Opus graffiti standing 171→172 is a separate counter. Prior map-check skips still hold: WOW-II 287/300/281 and WOW 136 are historical tips, not new +N.